I'm voting to close; this question is now several months old and is rather stale. Moreover, none of the answers so far are good.
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Kevin H. LinJun 3 '10 at 3:42

13

I don't know whether MO questions can get stale, as long as there are new MO users who haven't seen them before. Or are new users supposed to familiarize themselves with all previous questions?
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John StillwellJun 3 '10 at 4:38

2

Moreover, who is really capable of judging if "none of the answers is good" when talking about a soft-question (and be confident that he could convince the rest of the people about him being right)? I learned some things from these answers, and I am pretty sure that there probably is at least one person that found good any particular answer - namely, its poser!
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Jose BroxJul 22 '10 at 13:03

Or perhaps GTM 58, Koblitz, $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions, which features Fomenko's conception of the 3-adic unit disk. Or GTM 97, Koblitz, Introduction to Elliptic Curves and Modular Forms, with a Fomenko drawing depicting the family of elliptic curves that arises in the congruent number problem.
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Gerry MyersonJun 2 '10 at 4:21

Curiously, in the Russian translations of Koblitz's books, they have been transposed.
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Victor ProtsakJun 2 '10 at 6:00

Victor, the Russian translation of Koblitz's book on p-adic analysis has Fomenko's drawing of the 2-adic solenoid (both on the cover and inside the book), not a family of elliptic curves. That I checked directly. By a web search I find that the picture in the ell. curves book is the same in the translated volume as in the original one too.
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KConradJul 7 '10 at 23:01

Wow, this takes me back. <i>Not Knot</i> completely changed my impression of mathematical visualization and what it could do; to my mind it still sets the standard for expository mathematics video, and it's a shame that more aren't trying to reach for the bar it sets.
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Steven StadnickiJul 7 '10 at 0:14

Categorists have developed a symbolism that allows one quickly to visualize quite complicated facts by means of diagrams.

For me, this represents the fact that most, if not all of mathematics, is about structures and relations: even the simplest of them, when combined and interrelated, can give birth to fairly complex behaviour.

I really like combinatorics because of it's ability to formalize basic notions like comparison (being greater than, less than, or equal to some amount) and counting. For example here is a proof without words (from Ferrar's Diagram - Wikipedia) that the number of partitions of n objects with distinct odd parts is the same as the number of self-conjugate partitions. Math allows us to see patterns like this simply by moving colored dots around. How epic is that! To me, that's what combinatorics is all about.

Mathematics sometimes evokes emotion, as does music, and sometimes other forms of art. The strongest emotion I remember was the first time I saw Euler's Identity. So my image is just Euler's Identity written on a chalk board.